Unicorn family

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This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The unicorn family tempers out the unicorn comma (monzo: [-2 13 -8⟩, ratio: 1594323/1562500).

Unicorn

Unicorn divides the perfect fourth into eight equal parts, three for 10/9 and five for 6/5. Its ploidacot is omega-octacot.

Subgroup: 2.3.5

Comma list: 1594323/1562500

Mapping: [⟨1 2 3], ⟨0 -8 -13]]

mapping generators: ~2, ~250/243

Optimal tunings:

  • WE: ~2 = 1200.0889 ¢, ~250/243 = 62.4623 ¢
error map: ⟨+0.089 -1.476 +1.943]
  • CWE: ~2 = 1200.0000 ¢, ~250/243 = 62.4494 ¢
error map: ⟨0.000 -1.550 +1.844]

Optimal ET sequence: 19, 58, 77, 96, 173, 269

Badness (Sintel): 3.53

Septimal unicorn

The canonical extension to the 7-limit is by interpreting the generator as a very slightly flattened ~28/27, corresponding to tempering out 126/125, the starling comma, and 10976/10935, the hemimage comma. As it equates the perfect fourth with a stack of eight 28/27's, it also tempers out the octaphore comma in the 2.3.7 subgroup.

The hemifourth, reached by four generator steps, can be interpreted as ~15/13. This gives rise to a natural 2.3.5.7.13-subgroup extension where 196/195 and 676/675 vanish.

From 10976/10935 = (784/783)2⋅(841/840) we obtain an add-29 extension for free, so the generator triples as 28/27~29/28~30/29, and as 196/195 vanishes already, so does 729/728, leading to its S-expression-based comma list of {S26, S27, S28, S29}.

Experimentation shows we can find a reasonable mapping for prime 43 at -11 generator steps, tempering out 216/215. All other primes require either quite complex mappings (being significantly positive rather than negative) or require high error or both.

In all the subgroups above, the optimum is between 58edo and 77edo, though a notable tuning not appearing in the optimal ET sequences here is 96edo using the 96d val (with a 963 ¢ ~7/4 similar to that of meanpop), which sacrifices the accuracy of prime 7 in favour of a more accurate 5.

7-limit

Subgroup: 2.3.5.7

Comma list: 126/125, 10976/10935

Mapping: [⟨1 2 3 4], ⟨0 -8 -13 -23]]

Optimal tunings:

  • WE: ~2 = 1199.6949 ¢, ~28/27 = 62.2621 ¢
error map: ⟨-0.305 -0.662 +3.364 -2.074]
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.2996 ¢
error map: ⟨0.000 -0.352 +3.792 -1.717]

Optimal ET sequence: 19, 39d, 58, 77, 135c, 212c

Badness (Sintel): 1.04

2.3.5.7.13 subgroup

Subgroup: 2.3.5.7.13

Comma list: 126/125, 196/195, 676/675

Subgroup-val mapping: [⟨1 2 3 4 5], ⟨0 -8 -13 -23 -25]]

Optimal tunings:

  • WE: ~2 = 1199.7013 ¢, ~28/27 = 62.2785 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.3151 ¢

Optimal ET sequence: 19, 39df, 58, 77, 212cf

Badness (Sintel): 0.590

2.3.5.7.13.29 subgroup

Subgroup: 2.3.5.7.13.29

Comma list: 126/125, 196/195, 261/260, 377/375

Subgroup-val mapping: [⟨1 2 3 4 5 6], ⟨0 -8 -13 -23 -25 -22]]

Optimal tunings:

  • WE: ~2 = 1199.7715 ¢, ~28/27 = 62.2860 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.3141 ¢

Optimal ET sequence: 19, 39dfj, 58, 77, 212cf

Badness (Sintel): 0.487

Alicorn

Subgroup: 2.3.5.7.11

Comma list: 126/125, 540/539, 896/891

Mapping: [⟨1 2 3 4 3], ⟨0 -8 -13 -23 9]]

Optimal tunings:

  • WE: ~2 = 1198.6510 ¢, ~28/27 = 62.0316 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.1435 ¢

Optimal ET sequence: 19, 39d, 58

Badness (Sintel): 1.29

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 196/195, 676/675

Mapping: [⟨1 2 3 4 3 5], ⟨0 -8 -13 -23 9 -25]]

Optimal tunings:

  • WE: ~2 = 1198.6298 ¢, ~28/27 = 62.0480 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.1636 ¢

Optimal ET sequence: 19, 39df, 58

Badness (Sintel): 0.978

Camahueto

Subgroup: 2.3.5.7.11

Comma list: 126/125, 385/384, 10976/10935

Mapping: [⟨1 2 3 4 2], ⟨0 -8 -13 -23 28]]

Optimal tunings:

  • WE: ~2 = 1200.5186 ¢, ~28/27 = 62.4576 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.4252 ¢

Optimal ET sequence: 19, 58e, 77, 96d, 173d

Badness (Sintel): 2.18

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 385/384, 676/675

Mapping: [⟨1 2 3 4 2 5], ⟨0 -8 -13 -23 28 -25]]

Optimal tunings:

  • WE: ~2 = 1200.5004 ¢, ~28/27 = 62.4603 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.4277 ¢

Optimal ET sequence: 19, 77, 96d, 173d

Badness (Sintel): 1.49

Qilin

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 10976/10935

Mapping: [⟨1 2 3 4 6], ⟨0 -8 -13 -23 -49]]

Optimal tunings:

  • WE: ~2 = 1199.3865 ¢, ~28/27 = 62.1645 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.2199 ¢

Optimal ET sequence: 19e, …, 58, 135c, 193c

Badness (Sintel): 1.37

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 176/175, 196/195, 2200/2197

Mapping: [⟨1 2 3 4 6 5], ⟨0 -8 -13 -23 -49 -25]]

Optimal tunings:

  • WE: ~2 = 1199.2874 ¢, ~28/27 = 62.1601 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/27 = 62.2251 ¢

Optimal ET sequence: 19e, …, 58, 135c, 193cf

Badness (Sintel): 0.944

Monocerus

Subgroup: 2.3.5.7.11

Comma list: 126/125, 243/242, 5488/5445

Mapping: [⟨2 4 6 8 9], ⟨0 -8 -13 -23 -20]]

Optimal tunings:

  • WE: ~99/70 = 599.8223 ¢, ~28/27 = 62.2737 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~28/27 = 62.3180 ¢

Optimal ET sequence: 38d, 58, 96d, 154, 212ce

Badness (Sintel): 1.74

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 196/195, 243/242, 364/363

Mapping: [⟨2 4 6 8 9 10], ⟨0 -8 -13 -23 -20 -25]]

Optimal tunings:

  • WE: ~55/39 = 599.8267 ¢, ~28/27 = 62.2833 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~28/27 = 62.3262 ¢

Optimal ET sequence: 38df, 58, 96d, 154

Badness (Sintel): 1.19

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 196/195, 221/220, 243/242, 289/288

Mapping: [⟨2 4 6 8 9 10 9], ⟨0 -8 -13 -23 -20 -25 -8]]

Optimal tunings:

  • WE: ~17/12 = 600.0715 ¢, ~28/27 = 62.3767 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~28/27 = 62.3605 ¢

Optimal ET sequence: 38df, 58, 96d, 154

Badness (Sintel): 1.24

Rhinoceros

Rhinoceros tempers out the semaphoresma and the ragisma, identifying the generator as a ~21/20, rather than the 28/27 of septimal unicorn. It may be described as the 19 & 20c temperament. 19edo is a good tuning, though 39edo is a possible alternative.

Subgroup: 2.3.5.7

Comma list: 49/48, 4375/4374

Mapping: [⟨1 2 3 3], ⟨0 -8 -13 -4]]

Optimal tunings:

  • WE: ~2 = 1203.1031 ¢, ~21/20 = 63.0830 ¢
error map: ⟨+3.103 -0.412 +2.917 -11.848]
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 62.6687 ¢
error map: ⟨0.000 -3.305 -1.007 -19.501]

Optimal ET sequence: 1c, …, 18bcd, 19

Badness (Sintel): 2.07

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 100/99, 126/121

Mapping: [⟨1 2 3 3 4], ⟨0 -8 -13 -4 -10]]

Optimal tunings:

  • WE: ~2 = 1201.1458 ¢, ~21/20 = 62.9338 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 62.7783 ¢

Optimal ET sequence: 1ce, 18bcd, 19

Badness (Sintel): 1.96

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 78/77, 100/99, 126/121

Mapping: [⟨1 2 3 3 4 4], ⟨0 -8 -13 -4 -10 -6]]

Optimal tunings:

  • WE: ~2 = 1202.2047 ¢, ~21/20 = 63.1591 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/20 = 62.8727 ¢

Optimal ET sequence: 1ce, 18bcdf, 19

Badness (Sintel): 1.63

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